7.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{12675be8-6167-495b-a167-43b705b5ea5f-3_524_1310_808_292}
\end{figure}
Fig. 1 shows part of the curve \(C\) with equation \(y = \frac { 3 } { 2 } x ^ { 2 } - \frac { 1 } { 4 } x ^ { 3 }\).
The curve \(C\) touches the \(x\)-axis at the origin and passes through the point \(A ( p , 0 )\).
- Show that \(p = 6\).
- Find an equation of the tangent to \(C\) at \(A\).
The curve \(C\) has a maximum at the point \(P\).
- Find the \(x\)-coordinate of \(P\).
The shaded region \(R\), in Fig. 1, is bounded by \(C\) and the \(x\)-axis.
- Find the area of \(R\).